Mathematical Sciences: Supplement To: Ergodic Theory, Groupsand Geometry - Differential Geometry and Dynamical Systems by: Renato Feres

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Feres will study several problems in differential geometry and dynamical systems whose aim is to understand certain phenomena of rigidity of group actions. The solution to these problems could shed some light on a general conjecture of Zimmer's concerning actions of higher rank lattice groups and improve certain aspects of Gromov's theory of rigid transformation groups in the special case when such groups possess elements that define dynamical systems with hyperbolic behavior. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.